Press n or j to go to the next uncovered block, b, p or k for the previous block.
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import { numberToScreenX } from '../numberLineTicks'
import { primeColorRgba } from './primeColors'
import { smallestPrimeFactor } from './sieve'
import { decayingSin } from '../../shared/animationMath'
// --- Sieve phase timing ---
export interface SievePhase {
factor: number
startMs: number
durationMs: number
}
export const SIEVE_PHASES: SievePhase[] = [
{ factor: 2, startMs: 4000, durationMs: 5000 }, // seg 1: sweep 4000–9000, tail 9000–10200
{ factor: 3, startMs: 10200, durationMs: 3000 }, // seg 2: sweep 10200–13200, tail 13200–14400
{ factor: 5, startMs: 14400, durationMs: 1800 }, // seg 3a: sweep 14400–16200
{ factor: 7, startMs: 16600, durationMs: 1300 }, // seg 3b: sweep 16600–17900
{ factor: 11, startMs: 18300, durationMs: 800 }, // seg 3c: sweep 18300–19100, tail 19100–20300
]
export const CELEBRATION_START_MS = 20300
export const COMPOSITION_START_MS = CELEBRATION_START_MS + 5000 // 25300 — after celebration settles
/** The composite number used for the factorization reveal */
const COMPOSITION_EXAMPLE = 12
/** How long each factor's arc chain takes to draw (ms per arc) */
const COMPOSITION_ARC_STAGGER_MS = 300
/**
* Fixed maximum N for sweep calculations. All sweep progress, composite mark
* timing, and hopper position use this constant so they stay in sync with the
* viewport tracking in getSieveViewportState. Viewport-dependent bounds are
* only used for visibility culling (deciding which composites to draw).
*/
export const SWEEP_MAX_N = 130
// --- Per-composite animation ---
/** Timing constants for each composite's animation after being marked */
const FLASH_DURATION = 120
const SHAKE_DURATION = 130 // 120–250ms after mark
const FALL_DURATION = 350 // 250–600ms after mark
const ANIM_TOTAL = FLASH_DURATION + SHAKE_DURATION + FALL_DURATION // 600ms
interface CompositeAnimState {
factor: number // which prime factor marked it
markTimeMs: number // exact dwellElapsedMs when sweep reached it
}
// --- Easing helpers ---
function clamp01(x: number): number {
return Math.max(0, Math.min(1, x))
}
function easeInQuad(t: number): number {
return t * t
}
function easeOutQuint(t: number): number {
return 1 - (1 - t) ** 5
}
/** Per-factor sweep easing: slow start → fast finish */
function sweepEase(t: number, factor: number): number {
// Factors 2, 3: ease-in quad (t²) — 50% of time covers first 25% of range
// Factors 5, 7: gentler t^1.5 — fewer composites to show
const power = factor <= 3 ? 2 : 1.5
return t ** power
}
function easeOutCubic(t: number): number {
return 1 - (1 - t) ** 3
}
/** Decompose n into its prime factors, e.g. 12 → [2, 2, 3] */
function primeFactors(n: number): number[] {
const factors: number[] = []
let remaining = n
while (remaining > 1) {
const p = smallestPrimeFactor(remaining)
if (p <= 1) break
factors.push(p)
remaining /= p
}
return factors
}
/**
* Group prime factors into skip-counting chains for the composition reveal.
* e.g. 12 → [{ factor: 2, multiples: [2, 4, 6, 8, 10, 12] }, { factor: 3, multiples: [3, 6, 9, 12] }]
* Each chain shows the full skip-counting path from the factor up to N.
*/
function compositionChains(n: number): { factor: number; multiples: number[] }[] {
const factors = primeFactors(n)
const seen = new Set<number>()
const chains: { factor: number; multiples: number[] }[] = []
for (const f of factors) {
if (seen.has(f)) continue
seen.add(f)
const multiples: number[] = []
for (let m = f; m <= n; m += f) {
multiples.push(m)
}
chains.push({ factor: f, multiples })
}
return chains
}
// --- Sieve debug tuning (module-level, same pattern as goldenRatioDemo) ---
let sieveTrackingRange = 20
let sieveFollowHops = 15
export function getSieveTrackingRange(): number {
return sieveTrackingRange
}
export function setSieveTrackingRange(v: number): void {
sieveTrackingRange = Math.max(5, v)
}
export function getSieveFollowHops(): number {
return sieveFollowHops
}
export function setSieveFollowHops(v: number): void {
sieveFollowHops = Math.max(1, Math.round(v))
}
// --- Compute marked composites with exact mark times ---
function computeCompositeStates(
maxN: number,
dwellElapsedMs: number,
viewportRight?: number
): Map<number, CompositeAnimState> {
const composites = new Map<number, CompositeAnimState>()
for (const phase of SIEVE_PHASES) {
if (dwellElapsedMs < phase.startMs) break
const p = phase.factor
// Sweep starts from the factor itself (skip counting: 2, 4, 6, 8...)
// but only marks composites at p*2 and beyond (p itself is prime).
const sweepStart = p
const firstMultiple = p * 2
const sweepRange = maxN - sweepStart
if (sweepRange <= 0) continue
const linearProgress = clamp01((dwellElapsedMs - phase.startMs) / phase.durationMs)
const phaseProgress = sweepEase(linearProgress, p)
const maxReached = sweepStart + sweepRange * phaseProgress
for (let m = firstMultiple; m <= Math.min(maxN, maxReached); m += p) {
if (composites.has(m)) continue // already marked by earlier factor
// Compute exact time this composite was reached by the sweep
// Invert the easing: find the linear t such that sweepEase(t) = fraction
const fractionAlongSweep = (m - sweepStart) / sweepRange
// Invert power easing: t = fraction^(1/power)
const power = p <= 3 ? 2 : 1.5
const linearFraction = fractionAlongSweep ** (1 / power)
const markTimeMs = phase.startMs + linearFraction * phase.durationMs
composites.set(m, { factor: p, markTimeMs })
}
// Once the hopper has left the viewport, mark ALL remaining multiples
// of this factor as already fallen. This prevents un-eliminated composites
// from being visible when the viewport zooms out.
if (viewportRight !== undefined && maxReached > viewportRight) {
for (let m = firstMultiple; m <= maxN; m += p) {
if (composites.has(m)) continue
composites.set(m, { factor: p, markTimeMs: dwellElapsedMs - ANIM_TOTAL - 1 })
}
}
}
return composites
}
// --- Get active sweep info ---
function getActiveSweep(
dwellElapsedMs: number,
maxN: number
): { factor: number; sweepX: number } | null {
for (let i = SIEVE_PHASES.length - 1; i >= 0; i--) {
const phase = SIEVE_PHASES[i]
if (dwellElapsedMs < phase.startMs) continue
const linearProgress = clamp01((dwellElapsedMs - phase.startMs) / phase.durationMs)
if (linearProgress >= 1) continue
const progress = sweepEase(linearProgress, phase.factor)
const sweepStart = phase.factor
const sweepValue = sweepStart + (maxN - sweepStart) * progress
return { factor: phase.factor, sweepX: sweepValue }
}
return null
}
// --- Viewport keyframes for dynamic zoom/pan ---
export interface SieveViewport {
center: number
pixelsPerUnit: number
}
export interface SievePhaseViewports {
factor: number
start: SieveViewport // zoomed-in: first ~10 new composites visible
end: SieveViewport // zoomed-out: ~40 new composites visible
}
/**
* Pre-compute per-factor viewport keyframes based on the distribution of
* newly eliminated composites. For each factor:
* - `start`: zoomed in so the first ~10 new composites fill the screen
* - `end`: zoomed out so ~40 new composites are visible
*/
export function computeSieveViewports(cssWidth: number, maxN: number): SievePhaseViewports[] {
// Run the actual sieve to find which composites are NEW for each factor
const alreadyMarked = new Set<number>()
const result: SievePhaseViewports[] = []
for (const phase of SIEVE_PHASES) {
const p = phase.factor
const newComposites: number[] = []
for (let m = p * 2; m <= maxN; m += p) {
if (!alreadyMarked.has(m)) {
newComposites.push(m)
alreadyMarked.add(m)
}
}
if (newComposites.length === 0) {
// Fallback: shouldn't happen for factors 2,3,5,7 with maxN=120
result.push({
factor: p,
start: { center: p * 5, pixelsPerUnit: cssWidth / (p * 10 * 1.4) },
end: { center: 55, pixelsPerUnit: 5 },
})
continue
}
// 10th new composite → zoomed-in range
const nthZoomedIn = Math.min(10, newComposites.length) - 1
const zoomedInLast = newComposites[nthZoomedIn]
const zoomedInRange = zoomedInLast - p
const zoomedInCenter = p + zoomedInRange / 2
const zoomedInPpu = cssWidth / (zoomedInRange * 1.4)
// 40th new composite (or last) → zoomed-out range
const nthZoomedOut = Math.min(40, newComposites.length) - 1
const zoomedOutLast = newComposites[nthZoomedOut]
const zoomedOutRange = zoomedOutLast - p
const zoomedOutCenter = p + zoomedOutRange / 2
const zoomedOutPpu = cssWidth / (zoomedOutRange * 1.4)
result.push({
factor: p,
start: { center: zoomedInCenter, pixelsPerUnit: zoomedInPpu },
end: { center: zoomedOutCenter, pixelsPerUnit: zoomedOutPpu },
})
}
return result
}
/** Logarithmic interpolation for smooth zoom transitions */
function lerpLog(a: number, b: number, t: number): number {
const logA = Math.log(a)
const logB = Math.log(b)
return Math.exp(logA + (logB - logA) * t)
}
/** Linear interpolation */
function lerp(a: number, b: number, t: number): number {
return a + (b - a) * t
}
/** Interpolate between two viewports using linear center + logarithmic ppu */
function lerpViewport(a: SieveViewport, b: SieveViewport, t: number): SieveViewport {
return {
center: lerp(a.center, b.center, t),
pixelsPerUnit: lerpLog(a.pixelsPerUnit, b.pixelsPerUnit, t),
}
}
/**
* Clamp a sieve viewport so the visible range stays within bounds.
* Left edge gets a 10% margin so the origin isn't flush with the screen edge.
* Right edge stays at SWEEP_MAX_N to avoid showing un-sieved composites.
*/
function clampSieveViewport(vp: SieveViewport, cssWidth: number): SieveViewport {
const halfRange = cssWidth / (2 * vp.pixelsPerUnit)
let { center, pixelsPerUnit } = vp
// 10% of visible width as left margin (origin isn't pinned to screen edge)
const leftMargin = halfRange * 0.2 // 10% of full width = 20% of halfRange
// If the viewport is wider than the usable range, zoom in to fit
const usableRange = SWEEP_MAX_N + leftMargin
if (2 * halfRange > usableRange) {
pixelsPerUnit = cssWidth / usableRange
const newHalf = cssWidth / (2 * pixelsPerUnit)
center = -leftMargin + newHalf
return { center, pixelsPerUnit }
}
// Shift center so edges don't exceed boundaries
if (center + halfRange > SWEEP_MAX_N) center = SWEEP_MAX_N - halfRange
if (center - halfRange < -leftMargin) center = -leftMargin + halfRange
return { center, pixelsPerUnit }
}
/**
* Map virtual dwell time → interpolated viewport for the sieve animation.
*
* During each sweep phase: interpolate start → end using eased progress
* (same easing as the sweep itself, so slow sweep = zoomed in).
* During gaps/tails between phases: quick zoom-in to next start.
* During celebration: zoom out to wide view.
*
* All returned viewports are clamped so the visible range stays within
* [0, SWEEP_MAX_N], preventing un-sieved composites from appearing.
*/
export function getSieveViewportState(
virtualDwellMs: number,
keyframes: SievePhaseViewports[],
celebrationVp: SieveViewport,
cssWidth = 800,
maxN = 120
): SieveViewport | null {
if (keyframes.length === 0) return null
// --- Factor 2 hopper-tracking ---
// Follow the hopper closely for the first N hops, showing M integers
// around it so kids can see skip counting clearly. Tunable via debug panel.
const followUntilValue = 2 + sieveFollowHops * 2
const trackingPpu = cssWidth / (sieveTrackingRange * 1.4)
// Before first phase: hold at factor 2's tracking start (centered on 2)
const firstPhase = SIEVE_PHASES[0]
if (virtualDwellMs < firstPhase.startMs) {
return clampSieveViewport({ center: firstPhase.factor, pixelsPerUnit: trackingPpu }, cssWidth)
}
// During celebration → composition: multi-phase viewport transition
const lastPhase = SIEVE_PHASES[SIEVE_PHASES.length - 1]
const lastPhaseEnd = lastPhase.startMs + lastPhase.durationMs
if (virtualDwellMs >= CELEBRATION_START_MS) {
const celebDuration = 1500
const lastKf = keyframes[keyframes.length - 1]
// Composition reveal: zoom into the example number
if (virtualDwellMs >= COMPOSITION_START_MS) {
const compVp: SieveViewport = { center: 7, pixelsPerUnit: cssWidth / (16 * 1.4) }
const compTransitionMs = 1500
const compT = clamp01((virtualDwellMs - COMPOSITION_START_MS) / compTransitionMs)
const eased = easeOutCubic(compT)
return clampSieveViewport(lerpViewport(celebrationVp, compVp, eased), cssWidth)
}
// Celebration zoom
const celebT = clamp01((virtualDwellMs - CELEBRATION_START_MS) / celebDuration)
const eased = easeOutCubic(celebT)
return clampSieveViewport(lerpViewport(lastKf.end, celebrationVp, eased), cssWidth)
}
// Find which phase or gap we're in
for (let i = 0; i < SIEVE_PHASES.length; i++) {
const phase = SIEVE_PHASES[i]
const kf = keyframes[i]
const phaseEnd = phase.startMs + phase.durationMs
// During this sweep phase
if (virtualDwellMs >= phase.startMs && virtualDwellMs < phaseEnd) {
const linearT = clamp01((virtualDwellMs - phase.startMs) / phase.durationMs)
const easedT = sweepEase(linearT, phase.factor)
// Factor 2: track the hopper for the first 15 hops
if (phase.factor === 2) {
const sweepValue = phase.factor + (maxN - phase.factor) * easedT
const followEndEasedT = (followUntilValue - phase.factor) / (maxN - phase.factor)
if (easedT <= followEndEasedT) {
// Pure tracking: center on hopper, tight zoom
return clampSieveViewport({ center: sweepValue, pixelsPerUnit: trackingPpu }, cssWidth)
}
// Transition from tracking to regular keyframe viewport
const transitionRange = 0.15
const transitionT = clamp01((easedT - followEndEasedT) / transitionRange)
const eased = easeOutCubic(transitionT)
const trackVp: SieveViewport = { center: sweepValue, pixelsPerUnit: trackingPpu }
const regularVp = lerpViewport(kf.start, kf.end, easedT)
return clampSieveViewport(lerpViewport(trackVp, regularVp, eased), cssWidth)
}
return clampSieveViewport(lerpViewport(kf.start, kf.end, easedT), cssWidth)
}
// In a gap/tail after this phase but before the next
const nextPhase = SIEVE_PHASES[i + 1]
if (nextPhase && virtualDwellMs >= phaseEnd && virtualDwellMs < nextPhase.startMs) {
const gapDuration = nextPhase.startMs - phaseEnd
const gapT = clamp01((virtualDwellMs - phaseEnd) / gapDuration)
const eased = easeOutCubic(gapT)
const nextKf = keyframes[i + 1]
// By end of any phase (including factor 2's tracking), the viewport
// has fully blended to kf.end, so the gap transition is uniform.
return clampSieveViewport(lerpViewport(kf.end, nextKf.start, eased), cssWidth)
}
}
// After last phase but before celebration: hold at last end, zoom toward celebration
if (virtualDwellMs >= lastPhaseEnd && virtualDwellMs < CELEBRATION_START_MS) {
const tailDuration = CELEBRATION_START_MS - lastPhaseEnd
const tailT = clamp01((virtualDwellMs - lastPhaseEnd) / tailDuration)
const eased = easeOutCubic(tailT)
const lastKf = keyframes[keyframes.length - 1]
return clampSieveViewport(lerpViewport(lastKf.end, celebrationVp, eased), cssWidth)
}
return null
}
// --- Per-tick transforms for main renderer ---
export interface SieveTickTransform {
opacity: number // 0 = hidden, 1 = normal
offsetX: number // horizontal shake (px)
offsetY: number // vertical fall (px)
rotation: number // radians
}
/**
* Compute per-tick transforms for the main renderer during the sieve animation.
* Composites shake and fall off; primes are unaffected (not in the map).
*
* Animation is POSITION-BASED, not timestamp-based: progress is derived
* directly from how far the hopper has traveled past each composite
* (`hopsPast = (sweepValue - m) / factor`). This guarantees zero latency
* between the hopper landing on a number and its shake/fall starting,
* because they share the exact same sweepValue.
*/
export function computeSieveTickTransforms(
maxN: number,
dwellElapsedMs: number,
cssHeight: number,
viewportRight?: number
): Map<number, SieveTickTransform> {
const transforms = new Map<number, SieveTickTransform>()
const alreadyMarked = new Set<number>()
// Animation thresholds in "hops past" units
const SHAKE_HOPS = 0.3
const FALL_HOPS = 0.7
const TOTAL_HOPS = SHAKE_HOPS + FALL_HOPS // 1.0 hop to fully disappear
for (const phase of SIEVE_PHASES) {
if (dwellElapsedMs < phase.startMs) break
const p = phase.factor
const firstMultiple = p * 2
const sweepStart = p
const sweepRange = SWEEP_MAX_N - sweepStart
if (sweepRange <= 0) continue
const linearProgress = clamp01((dwellElapsedMs - phase.startMs) / phase.durationMs)
const sweepValue = sweepStart + sweepRange * sweepEase(linearProgress, p)
// Once hopper leaves viewport, all remaining multiples are instantly gone
const hopperOffScreen = viewportRight !== undefined && sweepValue > viewportRight
for (let m = firstMultiple; m <= maxN; m += p) {
if (alreadyMarked.has(m)) continue
// Hopper hasn't reached this composite yet
if (!hopperOffScreen && sweepValue < m) break // multiples are ascending
alreadyMarked.add(m)
// Off-screen bulk elimination: instantly hidden
if (hopperOffScreen && sweepValue < m) {
transforms.set(m, { opacity: 0, offsetX: 0, offsetY: 0, rotation: 0 })
continue
}
// How many hops past this composite the hopper has traveled
const hopsPast = (sweepValue - m) / p
let opacity = 1
let offsetX = 0
let offsetY = 0
let rotation = 0
if (hopsPast <= SHAKE_HOPS) {
// Shake: starts the instant the hopper lands
const shakeT = hopsPast / SHAKE_HOPS
offsetX = decayingSin(shakeT, 4, 2) * 3
} else if (hopsPast <= TOTAL_HOPS) {
// Fall: gravity drop
const fallT = (hopsPast - SHAKE_HOPS) / FALL_HOPS
const easedFall = easeInQuad(fallT)
const driftDirection = m % 2 === 0 ? 1 : -1
offsetX = driftDirection * easedFall * 8
offsetY = easedFall * (cssHeight * 0.8)
rotation = driftDirection * easedFall * 0.6
opacity = 0.6 * (1 - fallT)
} else {
// Fully hidden
opacity = 0
}
transforms.set(m, { opacity, offsetX, offsetY, rotation })
}
// Once this factor's sweep is complete, hide ALL its multiples up to maxN
// (covers composites beyond SWEEP_MAX_N visible when viewport zooms out)
if (linearProgress >= 1) {
for (let m = firstMultiple; m <= maxN; m += p) {
if (alreadyMarked.has(m)) continue
alreadyMarked.add(m)
transforms.set(m, { opacity: 0, offsetX: 0, offsetY: 0, rotation: 0 })
}
}
}
// Safety net: once all sweeps are complete, hide ANY remaining composite
// not caught by factors 2,3,5,7,11 (e.g. 169=13²)
const lastPhase = SIEVE_PHASES[SIEVE_PHASES.length - 1]
const allSweepsComplete = dwellElapsedMs >= lastPhase.startMs + lastPhase.durationMs
if (allSweepsComplete) {
for (let n = 4; n <= maxN; n++) {
if (transforms.has(n)) continue
if (smallestPrimeFactor(n) === n) continue // prime — leave alone
transforms.set(n, { opacity: 0, offsetX: 0, offsetY: 0, rotation: 0 })
}
}
// Composition reveal: fade the example composite back in as a ghost
if (dwellElapsedMs >= COMPOSITION_START_MS) {
const ghostT = clamp01((dwellElapsedMs - COMPOSITION_START_MS - 800) / 600) // delay + fade
if (ghostT > 0) {
transforms.set(COMPOSITION_EXAMPLE, {
opacity: ghostT * 0.5, // semi-transparent ghost
offsetX: 0,
offsetY: 0,
rotation: 0,
})
}
}
return transforms
}
// --- Main renderer ---
/**
* Render the animated Sieve of Eratosthenes overlay with "shake out" animation.
* Each composite flashes its number, shakes, then falls off the number line.
* Called each frame during the "ancient-trick" prime tour stop.
*/
export function renderSieveOverlay(
ctx: CanvasRenderingContext2D,
state: NumberLineState,
cssWidth: number,
cssHeight: number,
isDark: boolean,
dwellElapsedMs: number,
tourOpacity: number
): void {
if (tourOpacity <= 0) return
ctx.save()
ctx.globalAlpha = tourOpacity
const centerY = cssHeight / 2
// Visible range in number-line units (used only for visibility culling)
const EDGE_BUFFER = 5
const halfRange = cssWidth / (2 * state.pixelsPerUnit)
const leftValue = state.center - halfRange
const rightValue = state.center + halfRange
const visibleMin = Math.max(2, Math.floor(leftValue) - EDGE_BUFFER)
const visibleMax = Math.ceil(rightValue) + EDGE_BUFFER
// Compute all composite animation states using fixed sweep range
// (SWEEP_MAX_N keeps timing in sync with viewport tracking)
const composites = computeCompositeStates(SWEEP_MAX_N, dwellElapsedMs, rightValue)
// --- Layer 0: Celebration axis wash (dims composites drawn by main renderer) ---
if (dwellElapsedMs >= CELEBRATION_START_MS) {
const celebrationElapsed = dwellElapsedMs - CELEBRATION_START_MS
const washRamp = clamp01(celebrationElapsed / 800)
const washAlpha = 0.4 * washRamp
ctx.fillStyle = isDark ? `rgba(26, 26, 46, ${washAlpha})` : `rgba(248, 248, 248, ${washAlpha})`
ctx.fillRect(0, 0, cssWidth, cssHeight)
}
// --- Layer 1: Flash glow ring on composites as sweep reaches them ---
// The main renderer handles tick+label transforms (shake, fall, hide).
// The overlay only draws the glow marking effect during the flash phase.
for (const [value, anim] of composites) {
if (value < visibleMin || value > visibleMax) continue
const localTime = dwellElapsedMs - anim.markTimeMs
if (localTime < 0 || localTime > FLASH_DURATION) continue
const baseX = numberToScreenX(value, state.center, state.pixelsPerUnit, cssWidth)
const t = localTime / FLASH_DURATION
const glowRadius = 12 + 8 * easeOutQuint(t)
const glowAlpha = 0.5 + 0.3 * (1 - t)
const gradient = ctx.createRadialGradient(baseX, centerY, 0, baseX, centerY, glowRadius)
gradient.addColorStop(0, primeColorRgba(anim.factor, glowAlpha, isDark))
gradient.addColorStop(1, primeColorRgba(anim.factor, 0, isDark))
ctx.beginPath()
ctx.arc(baseX, centerY, glowRadius, 0, Math.PI * 2)
ctx.fillStyle = gradient
ctx.fill()
}
// --- Shared arc peak height for path + hopper ---
// Both Layer 2a (path arcs) and Layer 2b (hopper) use this so they align exactly.
function computeArcPeak(screenDist: number): number {
return Math.min(30, Math.max(8, Math.abs(screenDist) * 0.35))
}
// --- Layer 2a: Viewport-wide skip arcs (the "hopper path") ---
// Repeating arcs across the visible range at the current skip distance.
// During transitions between factors the skip distance morphs smoothly.
{
let skipDist: number | null = null
let arcColorFactor = 2
let arcAlpha = 1
// Determine effective skip distance + color for the current time
for (let i = 0; i < SIEVE_PHASES.length; i++) {
const phase = SIEVE_PHASES[i]
const phaseEnd = phase.startMs + phase.durationMs
// During a sweep phase
if (dwellElapsedMs >= phase.startMs && dwellElapsedMs < phaseEnd) {
skipDist = phase.factor
arcColorFactor = phase.factor
break
}
// In a gap/tail between phases: morph skip distance
const nextPhase = SIEVE_PHASES[i + 1]
if (nextPhase && dwellElapsedMs >= phaseEnd && dwellElapsedMs < nextPhase.startMs) {
const gapT = clamp01((dwellElapsedMs - phaseEnd) / (nextPhase.startMs - phaseEnd))
const eased = easeOutCubic(gapT)
skipDist = lerp(phase.factor, nextPhase.factor, eased)
arcColorFactor = gapT < 0.5 ? phase.factor : nextPhase.factor
break
}
}
// Before first phase: fade in arcs at factor 2 spacing
if (skipDist === null && dwellElapsedMs < SIEVE_PHASES[0].startMs) {
skipDist = SIEVE_PHASES[0].factor
arcColorFactor = SIEVE_PHASES[0].factor
arcAlpha = clamp01(dwellElapsedMs / 1500) // fade in over 1.5s
}
// After last phase: fade out during celebration
if (skipDist === null && dwellElapsedMs >= CELEBRATION_START_MS) {
arcAlpha = 0
}
// Between last phase end and celebration: hold last factor, start fading
const lastP = SIEVE_PHASES[SIEVE_PHASES.length - 1]
const lastEnd = lastP.startMs + lastP.durationMs
if (skipDist === null && dwellElapsedMs >= lastEnd && dwellElapsedMs < CELEBRATION_START_MS) {
skipDist = lastP.factor
arcColorFactor = lastP.factor
arcAlpha = 1 - clamp01((dwellElapsedMs - lastEnd) / (CELEBRATION_START_MS - lastEnd))
}
// Draw the repeating arcs
if (skipDist !== null && arcAlpha > 0.01) {
// Start from the leftmost visible skip boundary
const startN = Math.floor(leftValue / skipDist) * skipDist
const endN = rightValue + skipDist
for (let n = startN; n < endN; n += skipDist) {
const aFromSX = numberToScreenX(n, state.center, state.pixelsPerUnit, cssWidth)
const aToSX = numberToScreenX(n + skipDist, state.center, state.pixelsPerUnit, cssWidth)
if (aToSX < -50 || aFromSX > cssWidth + 50) continue
const arcPeak = computeArcPeak(aToSX - aFromSX)
ctx.beginPath()
ctx.moveTo(aFromSX, centerY)
ctx.quadraticCurveTo((aFromSX + aToSX) / 2, centerY - arcPeak, aToSX, centerY)
ctx.strokeStyle = primeColorRgba(arcColorFactor, 0.15 * arcAlpha, isDark)
ctx.lineWidth = 1.5
ctx.stroke()
}
}
}
// --- Layer 2b: Skip-counting hopper ---
// A dot that hops along the path arcs. Traces the exact same quadratic Bezier curve.
// The hopper "stomps" new composites (full arc) and "skims" already-gone ones (low arc).
{
let hopperPhase: SievePhase | null = null
for (let i = SIEVE_PHASES.length - 1; i >= 0; i--) {
const phase = SIEVE_PHASES[i]
if (dwellElapsedMs < phase.startMs) continue
const lp = clamp01((dwellElapsedMs - phase.startMs) / phase.durationMs)
if (lp >= 1) continue
hopperPhase = phase
break
}
if (hopperPhase) {
const p = hopperPhase.factor
const linearProgress = clamp01(
(dwellElapsedMs - hopperPhase.startMs) / hopperPhase.durationMs
)
const progress = sweepEase(linearProgress, p)
const sweepStart = p
const sweepRange = SWEEP_MAX_N - sweepStart
const sweepValue = sweepStart + sweepRange * progress
// Which two multiples of p are we between?
const fromMultiple = Math.max(p, Math.floor(sweepValue / p) * p)
const toMultiple = fromMultiple + p
const hopT = clamp01((sweepValue - fromMultiple) / p)
// Skim vs. stomp: is the landing target already eliminated by an earlier factor?
const targetAlreadyGone = toMultiple > p && smallestPrimeFactor(toMultiple) < p
const skimScale = targetAlreadyGone ? 0.12 : 1
// Convert to screen space
const fromSX = numberToScreenX(fromMultiple, state.center, state.pixelsPerUnit, cssWidth)
const toSX = numberToScreenX(toMultiple, state.center, state.pixelsPerUnit, cssWidth)
const screenDist = toSX - fromSX
// Arc height: match the path arcs' quadratic Bezier exactly.
// Quadratic Bezier with control point at (midX, centerY - arcPeak):
// y(t) = centerY - arcPeak * 2 * t * (1 - t)
// For skims, scale the peak down.
const arcPeak = computeArcPeak(screenDist) * skimScale
const arcHeight = arcPeak * 2 * hopT * (1 - hopT)
// Hopper screen position
const hopperSX = fromSX + hopT * screenDist
const hopperY = centerY - arcHeight
// Ghost label for already-gone composites: brief fade-in/out near landing
if (targetAlreadyGone && hopT > 0.4) {
const ghostAlpha = Math.sin(Math.PI * clamp01((hopT - 0.4) / 0.6)) * 0.55
if (ghostAlpha > 0.01) {
ctx.save()
ctx.globalAlpha = ghostAlpha * tourOpacity
ctx.font = 'bold 13px system-ui, sans-serif'
ctx.textAlign = 'center'
ctx.textBaseline = 'top'
ctx.fillStyle = isDark ? '#888' : '#999'
ctx.fillText(String(toMultiple), toSX, centerY + 5)
const textWidth = ctx.measureText(String(toMultiple)).width
ctx.beginPath()
ctx.moveTo(toSX - textWidth / 2 - 2, centerY + 12)
ctx.lineTo(toSX + textWidth / 2 + 2, centerY + 12)
ctx.strokeStyle = isDark ? '#888' : '#999'
ctx.lineWidth = 1.5
ctx.stroke()
ctx.restore()
}
}
// Hopper dot (smaller + translucent for skims)
const dotRadius = targetAlreadyGone ? 3.5 : 6
const dotAlpha = targetAlreadyGone ? 0.35 : 0.9
ctx.beginPath()
ctx.arc(hopperSX, hopperY, dotRadius, 0, Math.PI * 2)
ctx.fillStyle = primeColorRgba(p, dotAlpha, isDark)
ctx.fill()
// Glow around hopper
const glowRadius = dotRadius * 2.5
const glow = ctx.createRadialGradient(hopperSX, hopperY, 0, hopperSX, hopperY, glowRadius)
glow.addColorStop(0, primeColorRgba(p, targetAlreadyGone ? 0.2 : 0.35, isDark))
glow.addColorStop(1, primeColorRgba(p, 0, isDark))
ctx.beginPath()
ctx.arc(hopperSX, hopperY, glowRadius, 0, Math.PI * 2)
ctx.fillStyle = glow
ctx.fill()
}
}
// Factor spotlight glow (pulsing highlight on the prime factor being used)
for (const phase of SIEVE_PHASES) {
if (dwellElapsedMs < phase.startMs) break
const phaseProgress = clamp01((dwellElapsedMs - phase.startMs) / phase.durationMs)
if (phaseProgress >= 1) continue
const factorX = numberToScreenX(phase.factor, state.center, state.pixelsPerUnit, cssWidth)
const glowRadius = 20
const pulsePhase = (dwellElapsedMs / 300) % (Math.PI * 2)
const pulseAlpha = 0.2 + 0.1 * Math.sin(pulsePhase)
const gradient = ctx.createRadialGradient(factorX, centerY, 0, factorX, centerY, glowRadius)
gradient.addColorStop(0, primeColorRgba(phase.factor, pulseAlpha, isDark))
gradient.addColorStop(1, primeColorRgba(phase.factor, 0, isDark))
ctx.beginPath()
ctx.arc(factorX, centerY, glowRadius, 0, Math.PI * 2)
ctx.fillStyle = gradient
ctx.fill()
}
// --- Layer 3: Prime celebration (after all sweeps) ---
if (dwellElapsedMs >= CELEBRATION_START_MS) {
const celebrationElapsed = dwellElapsedMs - CELEBRATION_START_MS
const celebrationRamp = clamp01(celebrationElapsed / 800)
for (let n = visibleMin; n <= visibleMax; n++) {
if (n < 2) continue
if (smallestPrimeFactor(n) !== n) continue // not a prime
const sx = numberToScreenX(n, state.center, state.pixelsPerUnit, cssWidth)
// Pulsing radial glow
const glowRadius = 20
const phaseOffset = n * 0.07
const pulsePhase = dwellElapsedMs / 400 + phaseOffset
const pulseAlpha = (0.25 + 0.15 * Math.sin(pulsePhase)) * celebrationRamp
const gradient = ctx.createRadialGradient(sx, centerY, 0, sx, centerY, glowRadius)
gradient.addColorStop(0, primeColorRgba(n, pulseAlpha, isDark))
gradient.addColorStop(1, primeColorRgba(n, 0, isDark))
ctx.beginPath()
ctx.arc(sx, centerY, glowRadius, 0, Math.PI * 2)
ctx.fillStyle = gradient
ctx.fill()
// Prime labels are drawn by the main renderer — no overlay labels needed
}
}
// --- Layer 4: Composition reveal (after celebration) ---
// Shows skip-counting paths that reach the example composite, revealing its factorization.
if (dwellElapsedMs >= COMPOSITION_START_MS) {
const compElapsed = dwellElapsedMs - COMPOSITION_START_MS
const chains = compositionChains(COMPOSITION_EXAMPLE)
const factors = primeFactors(COMPOSITION_EXAMPLE)
// Viewport zoom-in transition (fade out celebration glow, fade in composition)
const compRamp = clamp01(compElapsed / 1000)
// Dim the celebration glows so composition arcs stand out
if (compRamp > 0) {
const dimAlpha = 0.3 * compRamp
ctx.fillStyle = isDark ? `rgba(26, 26, 46, ${dimAlpha})` : `rgba(248, 248, 248, ${dimAlpha})`
ctx.fillRect(0, 0, cssWidth, cssHeight)
}
// Draw skip-counting arc chains, staggered by factor
let arcTimeOffset = 500 // initial delay for viewport to settle
for (const chain of chains) {
const { factor, multiples } = chain
// Each arc in the chain appears sequentially
for (let i = 0; i < multiples.length - 1; i++) {
const arcStartMs = arcTimeOffset + i * COMPOSITION_ARC_STAGGER_MS
const arcT = clamp01((compElapsed - arcStartMs) / 400) // 400ms to draw each arc
if (arcT <= 0) continue
const fromN = multiples[i]
const toN = multiples[i + 1]
const fromSX = numberToScreenX(fromN, state.center, state.pixelsPerUnit, cssWidth)
const toSX = numberToScreenX(toN, state.center, state.pixelsPerUnit, cssWidth)
// Same arc shape as the hopper path
const arcPeak = computeArcPeak(toSX - fromSX)
// Draw the arc with growing alpha
const alpha = easeOutCubic(arcT) * 0.6
ctx.beginPath()
ctx.moveTo(fromSX, centerY)
ctx.quadraticCurveTo((fromSX + toSX) / 2, centerY - arcPeak, toSX, centerY)
ctx.strokeStyle = primeColorRgba(factor, alpha, isDark)
ctx.lineWidth = 2.5
ctx.stroke()
// Small dot at each landing point
if (arcT > 0.5) {
const dotAlpha = clamp01((arcT - 0.5) * 2) * 0.7
ctx.beginPath()
ctx.arc(toSX, centerY, 3, 0, Math.PI * 2)
ctx.fillStyle = primeColorRgba(factor, dotAlpha, isDark)
ctx.fill()
}
}
// Offset the next factor's chain so they appear sequentially
arcTimeOffset += multiples.length * COMPOSITION_ARC_STAGGER_MS + 400
}
// Factorization label: "2 × 2 × 3 = 12"
const labelDelayMs = arcTimeOffset + 200
const labelT = clamp01((compElapsed - labelDelayMs) / 600)
if (labelT > 0) {
const exampleSX = numberToScreenX(
COMPOSITION_EXAMPLE,
state.center,
state.pixelsPerUnit,
cssWidth
)
const labelAlpha = easeOutCubic(labelT)
// Build label like "2 × 2 × 3 = 12"
const labelStr = factors.join(' × ') + ' = ' + COMPOSITION_EXAMPLE
ctx.save()
ctx.globalAlpha = labelAlpha * tourOpacity
ctx.font = 'bold 16px system-ui, sans-serif'
ctx.textAlign = 'center'
ctx.textBaseline = 'bottom'
ctx.fillStyle = isDark ? '#e0e0e0' : '#333'
// Background pill for readability
const textWidth = ctx.measureText(labelStr).width
const pillPadX = 8
const pillPadY = 4
const pillY = centerY - 45
ctx.fillStyle = isDark ? 'rgba(26, 26, 46, 0.85)' : 'rgba(255, 255, 255, 0.85)'
const pillLeft = exampleSX - textWidth / 2 - pillPadX
const pillRight = exampleSX + textWidth / 2 + pillPadX
const pillTop = pillY - 16 - pillPadY
const pillBot = pillY + pillPadY
const pillRadius = 6
ctx.beginPath()
ctx.moveTo(pillLeft + pillRadius, pillTop)
ctx.lineTo(pillRight - pillRadius, pillTop)
ctx.quadraticCurveTo(pillRight, pillTop, pillRight, pillTop + pillRadius)
ctx.lineTo(pillRight, pillBot - pillRadius)
ctx.quadraticCurveTo(pillRight, pillBot, pillRight - pillRadius, pillBot)
ctx.lineTo(pillLeft + pillRadius, pillBot)
ctx.quadraticCurveTo(pillLeft, pillBot, pillLeft, pillBot - pillRadius)
ctx.lineTo(pillLeft, pillTop + pillRadius)
ctx.quadraticCurveTo(pillLeft, pillTop, pillLeft + pillRadius, pillTop)
ctx.closePath()
ctx.fill()
// Draw each factor in its prime color, operators in neutral
ctx.textBaseline = 'bottom'
let curX = exampleSX - textWidth / 2
for (let i = 0; i < factors.length; i++) {
const fStr = String(factors[i])
ctx.fillStyle = primeColorRgba(factors[i], 1, isDark)
ctx.fillText(fStr, curX + ctx.measureText(fStr).width / 2, pillY)
curX += ctx.measureText(fStr).width
if (i < factors.length - 1) {
ctx.fillStyle = isDark ? '#aaa' : '#666'
ctx.fillText(' × ', curX + ctx.measureText(' × ').width / 2, pillY)
curX += ctx.measureText(' × ').width
}
}
// " = 12"
const eqStr = ' = ' + COMPOSITION_EXAMPLE
ctx.fillStyle = isDark ? '#e0e0e0' : '#333'
ctx.fillText(eqStr, curX + ctx.measureText(eqStr).width / 2, pillY)
ctx.restore()
}
}
ctx.restore()
}
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